12/2/22
By Pythagorean theorem, \[x^2+y^2=r^2\]
\[\sin\theta=\frac{y}{r}\] \[\cos\theta=\frac{x}{r}\]
\[\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{y}{x}\] \[\csc\theta=\frac{1}{\sin\theta}=\frac{r}{y}\]
\[\sec\theta=\frac{1}{\cos\theta}=\frac{r}{x}\] \[\cos\theta=\frac{1}{\tan\theta}=\frac{x}{y}\]
Find the exact value of all six trigonometric functions of the angle \(A\).
\[ r=\sqrt{\left(12\right)^2+\left(-5\right)^2}=\sqrt{169}=13\]
\[ \sin A=\frac{y}{r}={\color{red}-\frac{5}{13}},\csc A={\color{red}-\frac{13}{5}} \]
\[\cos A=\frac{x}{r}={\color{red}\frac{12}{13}},~ \sec A={\color{red}\frac{13}{12}}\]
\[\tan A=\frac{x}{y}={\color{red}-\frac{5}{12}},~ \cot A={\color{red}-\frac{12}{5}}\]
Determine the exact value of \(\tan 120^\circ\) . (Decimal approximations will NOT be accepted.)
\(120^\circ\) is in Quadrant II.
- \(\tan(120^\circ) < 0\)
- \(ref(120^\circ)=180^\circ-120^\circ= 60^\circ\) \[\tan 120^\circ=-\tan(60^\circ)=-\sqrt{3}\]